Dredging and mining is shifting
to deeper waters. For dredging with TSHD’s the limit is around 150 m of water
depth, but for mining the water depth can be hundreds or even thousands of
meters. New technologies have to be developed or copied and adapted from the
offshore industry. At the DelftUniversity, Offshore
Engineering, students carry out research into many subjects related to moorings
and mooring systems, like:

The holding capacity of drag anchors
in sand.

The holding capacity of drag anchors
in clay.

The holding capacity and operations of
suction anchors.

The use of the catenary equation in
moorings.

The methodology about choosing the
right anchor for different purposes.

The methodology about choosing the
right anchor line for different purposes and conditions.

The methodology about designing an
anchoring system for FPSO’s.

The methodology about designing an
anchoring system for SPAR’s.

The methodology about designing an
anchoring system for Semi-Sub’s.

The results of the research can
also be found on: http://www.offshoreengineering.org.
In dredging and offshore anchors are used for positioning, but also for
operations (the cutter suction dredger). In all cases it is evitable that a
proper estimation of the holding capacity is very useful when designing the
application.

The holding capacity of anchors
depends on the digging depth, the soil mechanical properties end of course the
dimensions and the shape of the anchor. The digging depth also depends on the
soil mechanical properties and the shape of the anchor. Now the first question
is of course how is the digging depth related to the soil mechanical properties
and the shape of the anchor and the second question is, how does this relate to
the holding capacity.

By means of deriving the
equilibrium equations of motion of the anchor and applying the cutting
theories, the digging behavior of anchors can be simulated. The main challenges
are, how to model the shape of the anchor and how to apply the existing cutting
theories to this complex shape. This paper gives a first attempt to derive
equilibrium equations based on the cutting theory of Miedema 1987.

Keywords: Dredging, Anchors, Holding Capacity, Soil
Mechanics

INTRODUCTION

This paper is the result of assignments carried out by students for the
Offshore Moorings course of the MSc program Offshore Engineering of the DelftUniversity.
In this study an analyses is made of the penetration behavior of an anchor in
sand. The following
points are be taken into account.

• The geometry of the anchor and how to simplify it.

• What happens
when the anchor penetrates the soil?

• Which forces
will occur during penetration?

• How to solve
this mathematically?

First of all, the most
common anchors on the market are analyzed and a general anchor geometry will be
chosen. This chosen geometry will be simplified to a 2D geometry, which will be
realistic for a first analysis. After this step the penetration behavior of an
anchor will be described in different phases, such a way that it is clear and
easy to understand. Forces on the soil layer, fluke, shank and mooring line
forces will be defined and analyzed. The forces will be described as a function
of the geometries including the relevant variable angles.

The geometry of the anchor

While searching for the
best simplified anchor geometry, knowledge of the mostcommon anchors on the
market is needed. Vrijhof Anchors 2005 gives a good overview of the most common
anchors.Two types of anchors can be
considered, horizontal load anchors and vertical load anchors.The vertical load anchor
can withstand both horizontal and vertical mooring forces. The horizontal
anchor or drag embedment anchor can only resist the horizontal loads.The drag embedment
anchor is mainly used for catenary moorings, where the mooring line arrives the
seabed horizontally. The vertical load anchor is used in taut leg mooring
systems, where the mooring line arrives at a certain angle the seabed. A good starting-point for this case is to
analyze the horizontal load anchor, because this anchor is often used and will
form an adequate challenge.To
determine a simplified geometry of this horizontal anchor an actual figure of
this anchor is needed. In the figures 1
and 2, a sketch of the selected anchor can be found.

Figure
1: Examples of anchors

Figure
2: The geometry of anchors

The actual blade (fluke)
on the anchor, that will penetrate the soil is represented by the horizontal
part of the above given figure. The shank is represented by the other part of
the anchor. On the end of the anchor an anchor shackle can be found. A first simplification will be made by
modeling the anchor as a 2D model. Hereby all thecalculations will be made easier, but the geometry is still to
complex to determine all the forces. A second simplification can be
made by supposing the anchor as two straight lines as can be seen figure
3.This simplification is allowed,
because this is a conceptual (first) model. When all the forces and the
penetration curve of this simple model are known, a muchmore complicated model can be made. Figure 4 gives an overview of
drag embedded anchors.

Figure
3: The definition of fluke and shank

For the rest of the case a
few assumptions must be made. First, the type of soil will be sand, this way
the cohesion and adhesion effects can be neglected. As a next assumption it is considered that the anchor penetrates
the soil at a very low velocity. Therefore inertia and water tension can also
be neglected. Several constrains were also made to simplify the 3D force
analysis into a 2D analysis. This way several shear zones can be neglected. As
a final assumption, the force acting on the point of the fluke will be
neglected. This force is low considering a big anchor and will be fully
cancelled by the force perpendicular on the shank.

Figure
4: Typical drag embedded anchors (API 2005)

Soil resistance to
embedded anchor line

The
forces on the embedded anchor line can be seen as a separate system in the
anchor burial process and will be treated first. Up until 1989 the work
reported on embedded anchor chains was basically theoretical. Values of the
design parameters such as the effective chain width in sliding (Bs),
the effective chain width in bearing (Bb) and the bearing capacity
factor in clay (Nc) were suggested, but little experimental proof
confirmed their validity. With respect to general practice, the effective chain
widths can be expressed in terms of the nominal chain diameter (D):

(1)

(2)

Where
EWS and EWB are the parameters to express the effective widths in sliding and
bearing, respectively.

Vivatrat
et al. (1982) developed the analytical model of a chain inside soil by assuming
the chain length inside soil as a summation of short line segments and
expressing the equilibrium conditions of each segment. The effective width
parameters EWS and EWB proposed were 10 and 2.6, respectively and the use of a
value for Nc between 9 and 11 was suggested.

Yen
and Tofani (1984) performed laboratory measurements on sliding and bearing soil
resistances on a ¾ inch diameter stud-link chain in very soft silt. During
cutting and sliding tests, the maximum soil resistances were established to be
mobilized within a small movement of chain, even less than half a link. The EWS
parameter can be determined from laboratory tests and varies between 5.7 and
8.9. The Nc factor at a particular depth was found to be between 7.1
and 12.1, considering the EWB parameter to be 2.37.

Using
the finite segment approach, Dutta (1986, 1988) derived the nodal equilibrium
equations for chain segments and proposed a simple calculation method. This
study showed good agreement with the results obtained by the analytical method
used by Vivatrat et al. (1982).

Degenkamp
and Dutta (1989) derived a more accurate analytical model of embedded chain
under soil resistance and a simple calculation procedure. They used a soil
model to accurately predict the soil resistances to the chain inside soil and
estimated critical design parameters, such as effective widths of chain, based
on laboratory tests.

Assuming:
(1) Chain elements are inextensible; (2) due to the chain shift, the soil
medium suffers an undrained loading condition; (3) soil in the vicinity of the
chain reaches limit state of stress and thereby develops ultimate soil
resistances; and (4) the shear strength and weight of the soil over a chain
element are constant. the forces on a chain element are as presented in figure 5.

Figure 5: Force system on an
embedded anchor line

Assuming
is small, equilibrium
in tangential direction leads to:

(3)

and
by using the incremental integration approach, one can write:

(4)

Equilibrium
in normal direction ( is small) leads to:

(5)

or,
for discrete elements, applying the incremental approach:

(6)

The
entire embedded chain configuration can be assumed to be a summation of the
discrete chain elements. For each element, the value of and can be determined
using the known values of , , , and . To determine and , the tangential movement has been assumed to cause an
uncoupled sliding resistance, independent of the normal soil resistances. Using
this assumption, the frictional resistance is written as:

(7)

And
the normal soil resistance is written as:

(8)

The
value of is evaluated using the
formula given by Skempton (1951) for the ultimate net soil resistance of a
strip footing:

Where (with a maximum )

(9)

From
these equations, it can be seen that the accuracy of and are governed by the
factors (EWB x ) and (EWS x ) respectively. Extensive testing led to the next values:

For very soft (Su
= 5 kPa) clay:EWB = 2.5, EWS = 8.0

For firm (Su = 34
kPa) clay:EWB = 2.3,EWS = 7.2

Grote
(1993) used the exact same approach and values obtained by Degenkamp and Dutta
(1989) to determine the force distribution and geometric profile of the
embedded anchor line in his work to simulate the kinematic behavior of work
anchors.

Neubecker
and Randolph (1994, 1995) derived closed form expressions for both the load
development and chain profile to avoid the numerical solution by an incremental
integration technique used by Degenkamp and Dutta (1989) and simplify the
procedure. Their work corroborates the results found by Degenkamp and Dutta. For
the typical case where the chain angle at the seabed is zero, the expression
becomes:

(10)

The
expression for frictional development along the chain was derived as:

(11)

In
addition they found formulations for the embedded anchor line in sand using the
same approach, but changing the bearing capacity factor. For non-cohesive
soils, the bearing pressure may be expressed in
terms of a standard bearing-capacity factor as:

(12)

The kinematic
behavior of drag anchors in clay

Grote
(1993) derived a dynamic model to describe the anchor embedment. He described
the forces on the anchor as illustrated in figure 6. The forces on the anchor
consist of its weight, shear and normal forces on both fluke and shank and the
tension force from the anchor line pulling the anchor. The fluke force is calculated with a
transformation of the cutting formula of Miedema 1987. Grote states that the
force on the fluke depends on the depth and velocity of the anchor and a
constant, containing soil specific properties. This leads to a quantitative
formula of this force:

(13)

Figure 6: Forces on an embedded
anchor

The
second force on the fluke, , is a front force on the fluke caused by the bearing
capacity of the surrounding soil. This same force is present at the shank of
the anchor and called . They are both calculated with the formula for the bearing
resistance of a strip footing formulated by Terzaghi:

(14)

For
no free-draining soils as clay, it can be assumed that the internal friction
angle and . The second term is small compared to the last one, so
equation 16 can be simplified to:

(15)

is a shear force on
the shank. In clay ground the angle of internal friction is assumed zero while
the cohesion is not. The shear stress where failure occurs is then equal to the
cohesion, which is equal to the undrained shear strength.

(16)

And
finally, is the force on the
pad eye implied by the anchor line.

Steward
(1992) published methods to describe the kinematic behavior of drag anchors in
cohesive soils. These methods were simplified by Neubecker and Randolph (1995)
who formulated bearing capacity and moment equilibrium calculations utilizing
two fundamental anchor resistance parameters, and . The assumption that a drag anchor travels parallel to its
flukes is widely accepted. Therefore the authors expressed a geotechnical
resistance force acting on the anchor
parallel to the direction of the fluke, illustrated in figure 7, as:

(17)

Since
there will also be geotechnical forces normal to the fluke, the resultant
resistance force acting on the anchor
will make an angle with the fluke.

Figure 7:anchor parameter,

Therefore:

(18)

The
pad eye force can be determined as
the resultant of and the anchor weight . As the anchor embeds, the upper fluke surface and resultant
force will be at angles of and respectively to the
horizontal. Allowing for the offset angle between and , it then follows that:

(19)

Neubecker
and Randolph (1994, 1995) also developed an expression for the anchor chain
tension and angle at the anchor padeye assuming the chain angle at the seabed
is zero:

(20)

Every
anchor can be considered to have unique properties and that are independent
of the anchor size or the soil strength profile, and can be determined by
experimental modeling or comparison against published field data. The equations
described above can then be implemented into an incremental simulation as
follows:

1.Assume
an anchor fluke orientation and displace the
padeye horizontally an increment .

3.Calculate
the anchor resistance from the anchor
characteristics, anchor orientation and local soil strength.

4.Calculate
the chain angle using equation 20.

5.Calculate
the new fluke angle using equation 19 in
order to maintain equilibrium.

6.Displace
the padeye a further increment and loop to step 2.

By
adjusting the variables involved, the authors showed agreement with centrifuge
and full-scale tests.

Thorne
1998 developed a theory from geotechnical principals, without the use of any
site or anchor specific correlations. His predictions of the anchor movement showed
good agreement with nine full scale tests covering three different sites and
five anchor types. The equations used are based on the proposition that no
movement will occur until the soil forces acting parallel to the fluke are
overcome. The motion of the anchor results in the soil around the shank failing
in bearing capacity on the underside and in shearing on the base and sides,
exerting on the shank the maximum force of which the soil is capable. This is
also true for other elements which have to be dragged through the soil like
shackles, palms and stabilizers. These forces are calculated as:

(21)

where
and are the area and drag
factor for the “ith” component, acting at an angle to the plane of the
fluke on the anchor element (figure 8).

Figure 8: The anchor model

Figure 9: The force application
points

There
are three drag components: base bearing and skin adhesion of the base (acting
at right angles to the element), and skin adhesion on the sides (acting
parallel to the fluke). The drag factors are shape dependant and taken from
geotechnical research on soil forces on cylinders, flat plates, wedges and
strip footings. All forces are assumed to act at the centers of the respective
areas and the undrained shear strength is taken as that at the centre of area.
The forces of each element can be added to give the total drag force components
normal and parallel to the fluke and the moments about the fluke centre of
area:

(22)

(23)

(24)

Considering
an anchor with its fluke at an angle to the horizontal, the
centre of area of the fluke at a depth below the seabed and
the anchor chain at an angle to the horizontal
(figure 9), the equilibrium equations are:

.

(25)

(26)

(27)

To
solve this force system with the four unknowns , , , , one more equation is needed. Thorne used the closed form
expression given by Neubecker and Randolph 1995 for the anchor shackle tension,
, and the angle of the chain at the anchor to the horizontal,
:

(28)

The
proposed approach for progressive penetration is based on the pressure
distribution over a flat plate. An anchor fluke is considered, B long and L
wide, at some position acted upon by a normal force, , and a moment, , which results in the idealized contact pressure
distribution abcd and stress changes as the plate is moved a distance (figure 10).

Figure 10: The pressure distribution

These
stress changes can be considered as a loading on the plate consisting of an
equivalent normal force and moment acting at the centre of area.

(29)

(30)

Now
the incremental normal and angular deflections of the fluke after the movement are assumed to be
proportional to and with proportionality
constants and respectively. and are functions of the
elastic modulus of the soil, the plate size and the relative depth of the
plate. If the plate moves a total distance parallel to its plane,
there will be increment and the
resulting total deflections of the centre of area, and are:

(31)

(32)

To
allow estimation of the response of a real anchor fluke, an equivalent
rectangular fluke is used with the same centre of area and absolute first
moment of area. For a rectangle, the values of and are based on research
done by Lee (1962), Whitman and Rickart (1967), Butterfield and Bannerjee
(1971) and Rowe and Davis (1982) on deflection coefficients for a buried plate
(figure 11):

Figure 11: The deflection coefficients
for a buried plate

(33)

(34)

The
term takes account of the
aspect ratio of the rectangle, takes account of the
depth effect, is a reduction factor
to take account for post yield behavior and is the undrained
elastic modulus of the soil.

The
analysis of progressive movement now proceeds as follows. Assume the fluke
moving a distance , parallel to its plane, from an old position (n-1) to a new position n and rotating an angle from the initial angle
to an angle . The statics can be solved to find and for equilibrium. Now
using the progressive penetration approach as described above, it is possible
to estimate the angular displacement which would occur if
this combination of and were applied to the
plate. Repeating this for various angular displacements until will result in the
angle for this step. The
drag distance (horizontal movement), , of the anchor from position n-1 to n is then
calculated as:

(35)

and
the new depth for step n+1, , is calculated as below and the whole process is repeated.

(36)

A
value of was adopted for
calculation.

Finally
the American Petroleum Institute 2005 gives a graph for the holding capacity in
soft clay based on the ratio of the weight of the anchor and the holding
capacity. This graph is shown in figure 12, and gives lines for different types
of anchors. Of course this graph shows a worst case scenario based on soft
clay.

Figure 12: Anchor holding capacity
in soft clay (API 2005)

The kinematic
behavior of drag anchors in sand

Compared
to the behavior of an anchor in clay, the weight of the soil above the anchor
plays an important role in non cohesive soils like sand.

Le
Lievre and Tabatabaee 1981 proposed a limit equilibrium method. This method has
been shown to give reasonable predictions of the ultimate holding capacity of
drag anchors in sand, for a given depth of embedment. However, several
assumptions in the analytical procedure make it unsuitable for application to a
drag anchor during embedment, and hence the approach does not allow prediction
of the depth to which the anchor will embed, and thus the actual capacity. A
schematic representation of the failure mode and force system adopted by Le
Lievre and Tabatabaee is shown in figure 13.

For the
particular failure wedge angle, , calculate the side friction, SF. The side friction is
assumed to be the force required to overcome the lateral earth force
acting on the soil wedge area, hence:

(36)

From the force
polygon of the soil wedge plus anchor (figure 13), solve for the two
unknowns of chain tension, , and soil reaction, .

From either of
the other two force polygons of soil wedge only, solve for the values of
shank force, , and fluke force, .

Go back to step 1
and repeat with a different value of . Stop procedure after minimum chain tension is
calculated. This value is the best upper bound solution to the problem.

Grote
(1993) defined a force model shown in figure 10. His problem approach is the
same for clay as for sand grounds only the formula’s for the ground resistances
are different.

Figure 14: Forces on an embedded
anchor

The
fluke force is calculated with a
quantitative formula:

(37)

The
second force on the fluke, , is a front force on the fluke caused by the bearing
capacity of the surrounding soil. This same force is present at the shank of
the anchor and called. They are both calculated with the formula for the bearing
resistance of a strip footing formulated by Terzaghi:

(38)

where
,and are the bearing
capacity factors, the cohesion of the soil, the unit weight of the
soil, the width of the
footing (here fluke and shank) and the depth of the
bottom of the footing. For free-draining soils such as sand, the cohesion is thought to be zero.
The second term is small compared to the last one, so equation 38 can be
simplified to:

(39)

To
take account for the under pressure that can occur during anchor burial, the
author used a empirical equation which is a combination of the foundation
theory of Terzaghi and the cutting theory of Miedema set up by Becker et al.
1992:

(40)

where
is the pore under
pressure that follows from the cutting theory of Miedema.

is a shear force on
the shank. In sand ground the horizontal stress is assumed to be a function of
the vertical ground force:

(41)

Using
this relation, the formula used for the shear force is:

(42)

For
the value 2 was taken.
This value applies for piles driven into sand that densifies at the pile tip
due to the driving vibrations. And finally, is the force on the pad
eye implied by the anchor line.

Neubecker
and Randolph 1995 based their approach on the method of LeLievre and Tabatabaee
1981. They extended this method to incorporate:

·a more
realistic 3-dimensional failure pattern in the soil,

·a
force acting on the back of the fluke.

The
latter modification is particularly important at shallow penetrations, when the
bearing capacity of the anchor shank is insufficient to provide equilibrium (figure
15).

Figure 15: General force model

The
authors suggested that the force on the shank is dependent on its size and
shape and should be calculated from a bearing capacity viewpoint. Thus:

(43)

The
normal self-weight term () is omitted in equation 44, because it is assumed to have a
relatively small contribution and even the alone tends to over
predict the shank resistance.

The
3-dimensional failure mode

Dickin
1988 presented an overview of some of the various methods that have been
developed to evaluate the pullout resistance of a flat plate (figure 16).

It
was considered that the simple method of Majer (1955) consistently
underestimated the pullout capacity of the flat plate, while the model of
Vermeer and Sutjiadi (1985) gave predictions that compared well with
observations.

Neubecker
and Randolph incorporated the model of Vermeer and Sutjiadi into the drag
anchor problem as illustrated in figure 17.

Figure 17: 3-dimensional failure
mode

The
fluke can be thought of as being mapped onto the soil surface by the
displacement vector of the soil wedge. The failure planes are inclined at the
dilatation angle, , to the displacement vector so that, when they reach the
soil surface, the distance they are away from the fluke shadow is proportional
to the depth of the original point. This 3-dimensional soil failure mode is
still an idealized failure mode for the soil. However it does result in a more
realistic description of the failure surface. The area, , can be written as:

(44)

A
simplification is made in the calculation of the cross-sectional area, , in that a vertical slip surface is assumed behind the
fluke, rather than an inclined one. This is supposed to have a minor effect on
the balance of forces, as low active pressures are involved. However, no shear
force is assumed across this surface and as such the extra soil mass included
by this idealized failure mode counteracts the shear force that is neglected from
the realistic failure mode.

The
lateral extent of the failure wedge, , is determined from simple geometry.

(45)

Using
a pyramidical approximation for the sides of the wedge, the mobilized soil
mass, , is expressed as:

(46)

The
side friction, , that is to be used in the limit equilibrium equations is
obtained from Vermeer and Sutjiadi 1985.

(47)

Figure
18 and 19 show the system of forces with free body diagrams and force polygons
for the soil wedge only, the anchor only and the combined anchor-soil body.

Figure 18: Free body diagram

Figure 19: Force polygons

The
limit equilibrium calculation procedure is executed in the same way as that of
Le Lievre and Tabatabaee 1981 in that the chain tension is calculated for a
given failure wedge angle, , which is than varied until the chain tension reaches a
minimum. The procedure begins however, by initially examining force equilibrium
on the soil wedge only. The soil mass, , and the side friction, , are calculated using equations 46 and 47, respectively, for
the 3-dimensional soil wedge. The shank force, , is calculated from the standard bearing capacity
calculations from equation 43. The two unknown forces on the soil wedge, which
are the fluke force, , and the soil reaction, , can be calculated from horizontal and vertical force
equilibrium requirements. The force equilibrium of the anchor only is
considered now. The fluke force, , is defined from the previous equilibrium calculation, the
weight of the anchor is known and the shank force, , is calculated from equation 43. Again, there are only two
unknown forces left for solution by force equilibrium, namely the force on the
back of the fluke, , and the chain tension, . The failure wedge angle, , is then varied and the process repeated to obtain a minimum
upper bound estimate of . The solution of this force system still proceeds using
simple limit equilibrium methods, however, the equilibrium solution is applied
in a two step manner to arrive at a chain load and the introduction of an extra
unknown and an extra equation has resulted in a more realistic force system
acting on the anchor.

Finally
the American Petroleum Institute 2005 gives a graph for the holding capacity in
sand based on the ratio of the weight of the anchor and the holding capacity.
This graph is shown in figure 20, and gives lines for different types of
anchors.

Figure 20: Anchor holding capacity
in sand (API 2005)

Miedema et al 2006 derived a model based on 4
different penetration phases. The model is described below.

PENETRATION PHASES

Phase
1: No penetration

In this first situation the anchor lies on
the bed of soil and the fluke/shank angle will be considered as a minimum. When
pulling on the mooring line the anchor will scratch over the seabed, see figure
21. A bed of soil will be formed in front of the fluke and will give some
resistance. Because of this resistance, an angle κ will reach its maximum at a certain point.
At that certain point, the bed of soil in front of the fluke will give his
highest resistance and it will become easier to penetrate than scratching over
the seabed. When the assumption of a perfect sharp fluke point is made, the
point load can be neglected.

When the fluke starts to penetrate (figures
22 and 23), the cutting theory of Miedema 1987, can be used. Forces that will play a role in
the force balance are the fluke forces. When the angles on the fluke are
considered, a few assumption can be made. First of all the fluke/shank angle
κ will be constant and will have its maximum value. The internal friction
and the external friction angles are also constant. These parameters are only
depending on the material of the anchor and soil mechanical properties.

Figure 22: The fluke penetrating the soil in phase 2

Figure
23: The fluke in the soil

Figure 24: The forces on the soil layer

As discussed before the
cohesion, adhesion, inertial forces and water tension can be neglected. According
to figures 24 and 28, a force balance can be calculated.

The shear force
and the normal force are related according:

(48)

(49)

The grain forces
will be:

(50)

(51)

The weight of the
soil can be given as a force according:

(52)

Horizontal
equilibrium of forces:

(53)

Vertical
equilibrium of forces:

(54)

Figure 25: The forces on the fluke

The force K2 on the fluke is important to determine the
horizontal and vertical acting forces on the fluke (fig.25).

In this situation the fluke is completely
covered by sand and the shank will become an extra factor which will cause
penetration resistance, see figures 26 and 27. For the shank the cutting theory
of Miedema 1987, can’t be used. The strip footing theory as described in
Verruijt 2000, will be used for determining the shank resistance. The maximum
shank resistance acts when the complete shank is penetrated.

Figure 26: The shank penetrating the soil in phase 3

In this paragraph you will find the
modeling of the forces on the fluke and the shank. The influences of the angles
will be given. The cutting theory of Miedema 1987, is still valid for the fluke part of the
anchor forces. For determining the forces on the shank, the strip footing
theory, as described in Verruijt 2000, will be used. For phase 3, two shear zones are
taken into account. This will lead to a geometry as shown in figure 28.

Figure 27: The fluke and shank in
the soil

Figure
28: Forces on the soil layer

As discussed before; the cohesion, adhesion, inertial forces and water tension
can be neglected. For the figures 28 and 30, a force balance can be calculated.
The shear force and the normal force are related according:

The shear force
and the normal force are related according:

(58)

(59)

The grain forces
will be:

(60)

(61)

The weight of the
soil can be determined by using the geometry of figure 29, so the weight of the
soil will be:

For the determination of the forces on the
fluke and the shank, figure 30, two different theories will be used. For the
fluke the cutting theory of Miedema is valid, therefore forces on the soil
layer and on the fluke are the same as discussed in phase 2. For the forces on
the shank the strip footing theory can be used. This theory is based on the
fundamentals of Brinch Hansen and is a generalization of the Prantl theory.
This can be found in Verruijt 2000.

Figure
30: The forces on the anchor

To determine the friction Brinch Hansen
force P on the shank we can make use of:

(64)

Where c is cohesion and q is the external
load on the soil

Because c and q are zero in this case (no
cohesion and no external force on the soil), P will only be a function of the
soil weight part of the function, so:

(65)

Hereby is a correction factor for inclination
factors of the load. The factor is a shape factor for the shape of the load.

In this case only a load perpendicular to
the soil will be considered, so will be removed from the formula.

Inserting :

(66)

Now the friction
part of the shank has to be determined.

For the friction
of the shank, the next formula is valid:

Ffriction
= σn tan(δ) y h

(67)

It is possible now
to plot the results for P and Ffriction (see figure 31) then it is
possible to find out if the downward force of the fluke is big enough to pull
the shank through the seabed and further.

It is also
possible now to make a total force and moment balance, to predict the
trajectory of the anchor.

The fluke and the shank are completely
covered by sand (figures 32 and 33). When there is still no equilibrium, a part
of the mooring line will enter the soil. The
mooring line penetration will lead to an extra factor which will cause
penetration resistance. The anchor becomes stable when there is a balance
between the vertical and horizontal forces on the anchor part, which is covered
by sand.

In this phase you will find the modeling of
the forces on the fluke, the shank and the mooring line. The cutting theory of
Miedema is still valid for the fluke part of the anchor forces. For determining
the forces on the shank and the mooring line we will use the strip footing
theory as discussed in Verruijt.

Figure
33: Fluke, shank and mooring line in the soil

The soil layer
properties can be interpreted in a same way as described in phase 3. So this
way the function for G is still valid.

Figure
34: Forces on the fluke, shank and mooring line

To determine the
forces on the anchor for this situation the theory as discussed in phase 3 is
valid. For the mooring line forces (figures 34 and 35) we will also use the
Brinch Hansen theory as discussed in Verruijt 2000.

The penetration of
the mooring line causes resistance perpendicular to this line (penetration
resistance, see figure 36).

This effect is
noticeable in all soil conditions. The type of mooring line will determine the
value of this resistance. Think of a wire rope mooring line which penetrates
deeper (less resistance) than a chain mooring line.

During the penetration process
of the anchor, the resistance increases when depth increases, which is related
to the position of the anchor.

The mooring line
penetration can be described by the following geometry:

When looking at the point where the anchor
becomes stable, a force and moment balance can be made out of all the forces on
the anchor and mooring line. In fact this is the moment were the anchor reaches
his maximum holding capacity.

Figure
35: The forces on the anchor

Figure 36: The forces on the mooring line

Vertical equilibrium of forces:

(68)

Horizontal
equilibrium of forces:

(69)

Moment balance to
point A:

(70)

Conclusions
and recommendations

In the penetration
behavior the different forces and moments in all four phases can be described
with the theory dealt with, in this document. These forces and moments are a
function of the anchor geometry. The holding capacity of the anchor is
described as well as a function of the depth and the geometry. To predict the
trajectory of the anchor during the penetration, one has to find a relationship
between the different forces and moments on the anchor and the trajectory of
the anchor. The anchor trajectory will stop when the different forces are in
equilibrium, or when the pull force will be too high for that particular
anchor, at a certain depth. In the last case, the pull force necessary to
penetrate deeper in the soil, is higher then the maximum holding capacity of
that particular anchor at a certain depth. to that point the maximum holding
capacity is reached (see figure 37). When pulling further, the anchor will be
pulled out and looses his function.